Technical Report CS0920

We present a simple and unified approach for developing and analyzing
approximation algorithms for covering problems. We illustrate
this on approximation algorithms for the following problems:
Vertex Cover, Set Cover, Feedback Vertex Set, Generalized
Steiner Tree and related
problems.
The main idea can be phrased as follows:
iteratively, pay two dollars (at most) to reduce the total optimum by one
dollar (at least), so the rate of payment is no more
than twice the rate of the optimum reduction. This
implies a total payment (i.e., approximation cost )
$\leq$ twice the optimum cost.
Our main contribution is based on a formal
definition for covering problems, which includes all
the above fundamental problems and others. We
further extend the Bafna, Berman and Fujito
Local-Ratio theorem. This extension eventually
yields a short generic $r$-approximation algorithm
which can generate most known approximation
algorithms for most covering problems.
Using our technique, we present a linear time 2-approximation algorithm
for the Min Clique-Complement problem. The previous best result was a
4-approximation algorithm due to Hochbaum.
Another extension of the Local-Ratio theorem to
randomized algorithms gives a simple proof of Pitt's
randomized approximation for Vertex Cover. Using
this approach, we develop a modified greedy
algorithm, which for Vertex Cover, gives an expected
performance ratio $ \leq 2$.

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